z* = ½eiφfor 0≤φ≤2π

This equation is a parametric equation for the set of c values. It shows how the points
on the circle of radius 1/2 in the z* space map into the c space. For example, z*=½
maps into c=¼, z*=−½ maps into c=−3/4, z*=i½ maps into
c=½+i¼ and z*=−i½ maps into c=−½+i¼.

The plot below shows the full set of c values.

And there is the familiar cardioid shape that bounds the main body of the mandelbrot set,
as seen below.

The Limiting 2-Cycles

The iteration may approach a limit cycle rather than a limit point. For a two-period
cycle of z1* and z2* the values would have to satisfy the equations

z1* + z1*² = ¼eiφtherefore
c = −(1 + ¼eiφ)

This is the locus of a circle of radius ¼, the center of which is shifted one unit
to the left on the real axis. This is shown below in a diagram with the values of c corresponding
to limit points.

The Limiting 3-Cycles

The analysis for 3-cycles is more difficult but by same procedures as in the previous
it can be established that the boundaries between the stable and unstable cycles are circles of
radius 1/8. In general the boundaries between stable and unstable m-cycles are circles of
radius 1/2m. The diagram below shows the placement of circles having radii of
1/8, 1/16 and 1/32.

There are also easily discernible circles in the Mandelbrot set of radius 1/64.

The limit points z1*, z2* and z3* satisfy the equations

z3* = z2*² + c
z2* = z1*² + c
z1* = z3*² + c

The substitution of the equation for z1* into the equation for
z2*² and the substitution of that equation into the equation for
z3* produces a polynomial equation of degree 8.

There will be eight solutions
to this equation and two pairs of triples which satisfy the conditions for limiting 3-cycles
for each given value of c. The product of the eight solutions will be equal to the constant
term divided by the coefficient of the highest power of z*. This means that the product
of the solutions is equal to (c² + c)² + c.

In order to establish the conditions for stability of the 3-cycles consider the
deviations between zn+3 and z3*, zn+2 and z2*,
zn+1 and z1*, and zn and z3*. These deviations
satisfy the equations